A housing-demographic multi-layered nonlinear model to test regulation strategies
aa r X i v : . [ n li n . C G ] S e p A housing-demographic multi-layered nonlinear modelto test regulation strategies
Ram´on Huerta a , Fernando Corbacho b,c , Luis F. Lago-Fern´andez b,c, ∗ a Institute for Nonlinear Science, University of California, San Diego,La Jolla, CA 92093-0402 b Escuela Polit´ecnica Superior, Universidad Aut´onoma de Madrid,28049 Madrid (SPAIN) c Cognodata Consulting, Caracas 23, 28010 Madrid (SPAIN)October 25, 2018
Abstract
We propose a novel multi-layered nonlinear model that is able to capture and predictthe housing-demographic dynamics of the real-state market by simulating the transi-tions of owners among price-based house layers. This model allows us to determinewhich parameters are most effective to smoothen the severity of a potential marketcrisis. The International Monetary Fund (IMF) has issued severe warnings aboutthe current real-state bubble in the United States, the United Kingdom, Ireland, theNetherlands, Australia and Spain in the last years. Madrid (Spain), in particular, isan extreme case of this bubble. It is, therefore, an excellent test case to analyze hous-ing dynamics in the context of the empirical data provided by the Spanish
NationalInstitute of Statistics and other sources of data. The model is able to predict the meanhouse occupancy, and shows that i) the house market conditions in Madrid are unstablebut not critical; and ii) the regulation of the construction rate is more effective thaninterest rate changes. Our results indicate that to accommodate the construction rateto the total population of first-time buyers is the most effective way to maintain thesystem near equilibrium conditions. In addition, we show that to raise interest rateswill heavily affect the poorest housing bands of the population while the middle classlayers remain nearly unaffected. ∗ Corresponding author. E-mails: [email protected] (R. Huerta), [email protected] (F.Corbacho), [email protected] (Luis F. Lago-Fern´andez). Introduction
A crisis on the housing market could heavily reverse the current positive economical indi-cators [Kindleberger 2000, Garber 2000] and have a negative general impact in the worldeconomy, as shown in the Asian crisis in 1997 [Quigley 2001]. Real-state prices have in-creased more than fifty percent in Australia, Ireland, United Kingdom and Spain from 1997to 2004. These increases are hardly explainable in terms of economic fundamentals alone,even including record-low interest rates [Terrones et al. 2004]. Spain, in particular, holdshigh risk figures in terms of ratios of house prices to disposable income per worker (RPI)and house prices to rent (RPR). According to IMF calculations Spain has 3 .
6, 3 .
6, 2 .
5, 2 . . .
42, 3 .
32, 1 .
9, 1 .
8, and 1 .
28 times the RPR of Germany, Japan, France, United Statesand United Kingdom, respectively [Terrones et al. 2004]. The RPI index has been proved tobe a good reference of the distance to market equilibrium [Malpezzi 1999]. Therefore, Spainis a valid scenario to analyze the conditions to understand and control a housing market bymeans of nonlinear dynamical models.In this paper we develop a mean field model of the house occupancy derived from astochastic process as previously used in the study of epidemic dynamics [Bailey 1975]. Thisnonlinear model is employed as a tool to ascertain the possibility of sudden changes in thedynamics as a function of the control parameters. The model is based on dissecting thehouse population in several groups or layers. Each layer has a given number of houses,which can be either occupied or empty. Every family occupying a house in a given layer hasa certain probability rate to migrate to any other layer. In addition, there exists a base poolof nonowners, which can jump to the housing layers at a certain probability rate. Thus themodel captures the dynamics of the mean migrations between housing layers.As a particular case, we analyze the house market in the city of Madrid in the frameworkof the model. The parameters of the model have been obtained from different sources suchas the Spanish National Institute of Statistics, the Spanish Ministry, and some valuationcompanies [INE 1991, INE 2002, INE 2004, Sociedad de Tasaci´on S.A. 2004].Our main results may be summarized as follows: • The model is able to predict the occupancy levels in 2001 given the parameters obtainedfrom 1991 data. This is a good indication that the mean description is able to modelthe real dynamics. • The model shows that the house market for the city of Madrid asymptotically evolvesto an out-of-equilibrium condition. It is rather worrisome that the new housing unitscannot be replenished by first-time buyers. • Critical phenomena do not exist in the context of this model. Only smooth changescan be expected. 2
A sudden increase in the interest rate will seriously affect the occupancy level of thelowest layer of the market, i.e. the poorest housing unit sector, while the middle layersremain nearly unaffected. • According to the model, the most effective way to control the out-of-equilibrium situ-ation of the Madrid house market is to down regulate the construction rate.There is a large body of research in the housing market [Green and Malpezzi 2003,Malpezzi 1990], most of the analyses aiming at predicting house prices. We do not in-tend to estimate prices, our modeling efforts fit better with housing-demographic models[Mankiw and Weil 1989, Crone and Mills 1991]. The novelty of our approach lies on build-ing a multi-layered nonlinear dynamical model that includes family migrations across layersof housing units. The whole population is separated in non-overlapping housing bands, whichare estimated from census data. Our model neglects the random component of the stochasticprocess, because we use a mean field approach under the assumption of random mixing. Therandom mixing approximation basically states that any family can, in principle, uniformlyaccess any house. This random mixing approximation allows to obtain a set of ordinarydifferential equations (ODEs) derived from a stochastic process. The formalism that we usehere sets us apart from previous approaches.
In this section we provide an overall description of the non-linear dynamical model forthe house occupancy. The model equations will be explicitly derived in the next section.We divide the total housing population into different price bands or layers, and model thetransitions of owners among these layers. Note that, in this paper, we only use the price asa tool to determine the different house bands. Let us assume that there are N layers layers.We define N i ( t ) as the total number of houses, or units, in layer i at time t . Each of thelayers has a certain number of occupied houses, O i ( t ). These are defined as houses occupiedby owners. In addition to the house price bands, we consider a source of new buyers, whichwe call the base pool . It accounts for non-emancipated people, immigrants, and rented units.A group of people living in the same house will be called a family . The family is the basicpeople unit in our model, and so we are modeling family jumps among house layers. Notethat the number of families in a given layer i equals the number of occupied houses in thatlayer, O i ( t ). The concept of family applies to non-emancipated people and immigrants aswell. However, in these cases information concerning the total number of single individualsis more frequently available in census databases, so we must apply a correction factor whencalculating the number of families (Σ( t )) in the base pool.The migrations among house layers are modeled in terms of transition probabilities. Wedenote by µ ij the probability rate for a family in house layer j to move to house layer i .Equivalently, we denote the probability rate to move from the base pool to any of the houselayers i by η i . Finally, there is a probability for any family to disappear, or die, leaving its3ouse empty. We call this probability the death rate , λ i , which we assume to be constantfor each layer i . Figure 1A shows an overall scheme of the model; figure 1B displays a list ofthe variables and parameters involved.With all the above ingredients we pose a set of non-linear coupled equations for themean occupancy of each level: o i ( t ) = O i ( t ) /N i ( t ). The model equations are derived fromthe mean field approximation to a stochastic process, as used in epidemic dynamics. Detailsare provided in the next section. µ N (t)
Non−emancipated units+Rented units+Immigrant units1
N (t) LE V EL S η A (t) Σ B O i ( t ) Total number of occupied housing units inlayer i . N i ( t ) Total number of housing units in layer i . o i ( t ) O i /N i . η i Probability rate to jump from the base pool tolayer i . µ ij Probability rate to move from layer j to layer i .Σ( t ) Number of families in the base pool. λ i Instantaneous family death rate in layer i . Figure 1: (A) Explanatory figure of the model and the main parameters required in our housing-demographic model. Each layer represents a price housing band. The dark circles represent occupiedhousing units. Σ( t ) represents the pool of units that can enter any of the vacant sites on any of the layerswith some probability rate η . Each of the layers have a total number of housing units N i ( t ). The transitionprobability rate µ j denotes the probability of having transition from layer j into layer 3. Note that weomitted the rest of the transitions for the sake of the readability of the figure. (B) Glossary of variables andparameters of the nonlinear dynamical model. The model proposed here is similar in derivation to the ones used in epidemic modeling[Bailey 1975, Huerta and Tsimring 2002] where the stochastic epidemic process is reducedto a set of ordinary differential equations (ODEs). These ODEs capture quite faithfully thebehavior of the stochastic process behind epidemics. The main advantage of this approachis that the complexity of the process is simplified to a formalism that allows an easierunderstanding of the qualitative behavior. The parameters can be easily related to the endresult of the stochastic process. In most cases the ODEs match well the stochastic process,although there are some others where the ODE description fails, for example, for finite-sizeeffects. Overall, the ODE description is a very good framework to gain understanding thatcan complement very well stochastic modeling. Our main contribution is to bring theseconvenient tools to housing-demographic modeling.To model the dynamics across housing layers we use two possible states for any house:occupied and empty. The total number of houses in each layer i , N i ( t ), evolves in time4ccording to a function estimated from the census data. Given the number of occupied unitsin layer i , O i ( t ), and the transition probability rate from layer j to layer i , µ ij , the probabilitythat a family jumps from layer j to layer i in the time interval dt is µ ij { − O i ( t ) /N i ( t ) } dt .This probability already assumes that a family can only occupy an empty house in layer i .This is equivalent to the random mixing approximation widely used in epidemic modeling[Bailey 1975]. The net flow into layer i is the difference between the number of incomingand outgoing families in the time interval dt : F i = N layers X j =1 " O j ( t ) µ ij ( − O i ( t ) N i ( t ) ) − O i ( t ) µ ji ( − O j ( t ) N j ( t ) ) dt This flow equals the variation in the number of occupied states in layer i during the timeinterval dt , so we can write the following set of ordinary differential equations (ODEs) forthe layer occupancy: dO i ( t ) dt = N layers X j =1 " µ ij O j ( t ) ( − O i ( t ) N i ( t ) ) − µ ji O i ( t ) ( − O j ( t ) N j ( t ) ) It is critical to include the dynamical contribution of the base pool. The probabilityfor a family to jump from the base pool to the layer i in the time interval dt is given by η i { − ( O i ( t ) /N i ( t )) } dt . Then, the net change in layer i due to the flow from the base poolis simply Σ( t ) η i { − ( O i ( t ) /N i ( t )) } dt . Finally, we will consider a family death rate for eachlayer, λ i , which contributes to the variation in layer occupancy with the term − λ i O i ( t ) dt .The final model equations can be written as: dO i dt = η i − O i N i ( t ) ! Σ( t ) − λ i O i + N layers X j =1 " µ ij O j ( − O i N i ( t ) ) − µ ji O i ( − O j N j ( t ) ) (1)Since we plan to analyze the asymptotic behavior of the equations, we define a new variable, o i ( t ) ≡ O i ( t ) /N i ( t ), which is the normalized occupancy level, bounded between 0 and 1. Wecan rewrite the set of equations 1 in terms of the normalized occupancy as: do i dt = η i (1 − o i ) Σ( t ) N i ( t ) − o i ddt log N i ( t ) − λ i o i + N layers X j =1 µ ij o j (1 − o i ) N j ( t ) N i ( t ) − µ ji o i (1 − o j ) ! (2)There are three terms in these ODEs with explicit dependence on time. The first one isthe drive from the base pool to saturation levels. If Σ( t ) grows faster than single layers do,then all layers will saturate. The third term with explicit dependence on t is multiplied by N j ( t ) /N i ( t ). This term implies that, if the size of layer j grows much faster than layer i , inthe asymptotic limit the layer i will be totally full, with a huge demand in that layer thatwill quickly change the band location in the whole distribution of layers.5 Test case: the city of Madrid
Madrid is a particularly extreme case of the real-state bubble in Spain [Terrones et al. 2004].This fact, together with the availability of data to estimate model parameters, makes Madridan interesting test case to be analyzed in the context of our mean field model. The two mainsources of data used to feed the model parameters are the Spanish National Institute ofStatistics (INE) and the Spanish Ministry, but we have also used data provided by valuationcompanies and real-state internet sites. First we will provide an overall description of themodel parameter estimation in section 4.1. Then, in section 4.2, we will use these parametersin the model equations (2) to understand the implications of current market conditions.
First of all we must determine the price layers. Figure 2A represents the distribution of houseprices in the city of Madrid in 1991 (data from [INE 1991, Sociedad de Tasaci´on S.A. 2004]).This distribution provides the total number of houses per layer, N i , using 7 layers thataccount for the 99% of the total number of houses. The number of occupied houses ineach layer, O i , is also provided by [INE 1991, Sociedad de Tasaci´on S.A. 2004]. To choosethe price size of each layer we find a compromise between two opposing criteria: i) themaximum number of layers in order to have a detailed distribution of the housing stock; andii) the widest price size per band such that the transition probability rates between layers arenot very small when normalized to the integration time scale. This second criterion intendsto avoid finite size effect problems.The time evolution of the number of houses per layer, N i ( t ), is hardly available in publicdatabases. Nevertheless, as shown in fig. 2B, the total population of houses is available atfive different years since 1970 [INE 2004]. In that figure we can see that the total populationof houses is very well fit by an exponential function of time. We will assume that the shapeof the layers distribution is time invariant, and will apply the same exponential growth toall layers. This assumption is supported by two facts: (i) the evidence of self-regulation ofeach of the layers as shown in [Linneman 1986] for the city of Philadelphia, i.e. , undervaluedhouses compared to similar type of houses get more appreciated than the average; and (ii)the similarity of price distributions in the cities of Pitt County (North Carolina) [Bin 2004]and Madrid (figure 2A).The number of families in the base pool (Σ( t )), i.e. , the subset of families that do notown a property, is estimated from the INE databases as the sum of: (i) the number of peoplethat are not emancipated within the range of age where people usually emancipate (datafrom [INE 2002, OBJOVI 2004]); (ii) the number of families that rent an apartment (dataprovided by [INE 2004]); and (iii) the number of incoming families due to the immigration(data from [INE 2004]). These statistics do not provide family units, but they report singlepeople data instead. So a headship factor of 2 has been applied when calculating Σ( t ).In fig. 2C we can see the time evolution of the three subsets that conform the base pool.The two main contributions to the pool have been decreasing in the last few years. On the6ther hand the immigrant population has undergone a sharp increase. However, in contrastto the popular view, this subset of the population is not officially large, and it might justcompensate for the negative tendency in the base pool size time evolution. In contrast tothe total number of houses (fig. 2B), there is no exponential growth of Σ( t ). In fact, we willassume it is a constant. This situation, if maintained, would lead the system to completedepletion as we will discuss bellow in the context of the model.Other important parameters to be estimated are the transition probability rates fromthe base pool to the different layers, η i , and the transition probability rates among layers, µ ij . The average transition probability rate from the base pool to any of the layers, ¯ η , canbe estimated from the total number of new occupancies in 1981, 1991 and 2001, as follows.The total number of occupied houses in 1981 was 699 , ,
444 and in 2001was 908 ,
790 [INE 2004]. Therefore the occupancy rate from 1981 to 1991 was 8 ,
989 newoccupancies per year, and from 1981 to 1991 it was 11 ,
935 new occupancies per year. Thebase pool population in 1981 is estimated in 568 ,
300 families (we have data for the non-emancipated population since 1986 and we apply a headship per future household of 2), in1991 it was 532 ,
717 and in 2001 it was 543 , .
016 year − from 1981 to 1991,and 0 .
022 year − from 1991 to 2001. This rate has been increasing in the last few years,maybe due to the decrease in the interest rates.To fit the probabilities µ ij , we assume that the occupancy levels in 1991 are stationary,and search for the values of µ ij that provide these levels at equilibrium. The available datado not permit direct calculation of these probabilities, nevertheless it is possible to estimatetheir average by extrapolating the number of housing transactions in Spain (provided byanalysis office by [BBVA 2003] and [OBJOVI 2004]) to the city of Madrid. If we take thistotal number of transactions and discount the probability of transition from the base pooland the estimation of housing transactions due to investments, we obtain ¯ µ = 0 .
052 year − .This basically states that the time it takes a family to change to a new house is 20 years onaverage for the city of Madrid. This is imposed as a constraint in the following calculations.Therefore, we will assume that the occupancy levels are slightly off from the equilibriumstate and that the growth of the population is compensated with the growth of the totalnumber of housing units. The occupancy level of each of the 7 layers at 1991 is given byˆ o = (0 . , . , . , . , . , . , . o and the model equilibrium solution o . To reduce the search space we apply two constraints:(i) the average value of µ ij is, as stated above, ¯ µ = 0 .
052 year − ; and (ii) the derivativebetween close parameters is as small as possible. A total of 50 different simulations wererun, the outcomes of three of them are shown in figure 3. Although the specific values ofthe probabilities do not match for the three simulations, the shape and tendency maintainqualitative agreement. We use their average value when solving the model equations.The last important estimation in our model is the family death rate for each layer. Wehave data of the absolute number of deaths per age and the age distributions per housinglayer [INE 2004]. With these data, assuming a family is composed of two people very close7n age, we can calculate the instantaneous family death rate per layer (fig. 2D). We can seethat the lower layers have a higher death rate, because the average age of families in theselayers is higher. This can be explained by the slow integration of the old houses into thelower price layers. On the other hand, newly constructed houses tend to fill higher layers. Price Range (x 1000 Euros) N u m be r o f H ou s i ng U n i t s (DATA OBTAINED IN 1991) Ln ( o f hou s i ng un i t s i n X ) Price Range (x 1000 Euros) λ ( y ea r − ) NU M BE R O F UN I T S RENTING UNITSINMIGRATION UNITSNONEMANCIPATED UNITS
BA C D
Figure 2: (A) Distribution of house prices in the city of Madrid (Spain) during 1991, extracted from dataprovided by [INE 1991] and [Sociedad de Tasaci´on S.A. 2004]). (B) Fits to the logarithm of the total number(circles), primary (squares), secondary (triangles), and empty (stars) houses. The data is provided for only5 points in time [INE 2004]. The fits yield the following slopes: 0 . ± .
001 year − (total), 0 . ± . − (primary), 0 . ± .
001 year − (secondary), and 0 . ± .
003 year − (empty). In all the cases thecorrelation factor of the linear regression is greater than 0 .
98. (C) Time evolution of the different subsets ofthe base pool: non-emancipated families (stars), immigrant families (squares), and renting families (circles).(D) Instantaneous death rate per layer, λ i . Parameters estimated from 1991 data. For the problem under analysis, and taking into account the exponential growth of the totalnumber of houses for the city of Madrid, the model equations (2) can be rewritten as: do i dt = η i (1 − o i ) Σˆ N i exp k t − o i ( k + λ i ) + N layers X j =1 µ ij o j (1 − o i ) ˆ N j ˆ N i − µ ji o i (1 − o j ) ! (3)8here k is the constant of the exponential growth fitted in figure 2B, and ˆ N i is the totalnumber of houses in layer i in 1991 (figure 2A). Integrating the equations with the parametersof previous section, we can make predictions on the mean occupancy level. In particular, thedecreasing trend of the occupancy levels from 1991 to 2001 is well predicted by the modelusing data from 1971 to 1991 (see figure 4A). l a y e r l a y e r l a y e r l a y e r l a y e r l a y e r t r an s i t i on p r obab ili t i e s f r o m l a y e r destination layer poo l Figure 3:
Solutions of three different runs of the genetic algorithm used to estimate the transition proba-bilities. These plots show the values of the transition probabilities from any of the layers including the basepool (rows, y axis) to any of the 7 property layers ( x axis) in ten years. To get the probabilities in one yearthe values must be divided by 10. The standard deviation with respect to the target value ˆ o is in all thecases less than 10 − . A sufficient condition to keep the system asymptotically away from 0 (empty) or 1 (full)is the following N ( t ) = N (0) + Z t X i η i ! − ¯ o ¯ o e ¯ λ τ Σ( τ ) dτ ! e − ¯ λ t , (4)where ¯ o is the mean occupancy level, and ¯ λ is the mean instantaneous household death rate.If the time evolution of the base population is constant, as it appears according to figure 2C,then N ( t ) ∝ exp ( − ¯ λt ). The optimal equilibrium occupancy levels in the housing market isout of the scope of this paper. Nevertheless it is certainly desirable to steer the system toa equilibrium condition whose asymptotic state is far from the 1-0 extremes. Equation (4)9ntends to provide a simple framework for policy makers to regulate the system in a smoothway from a housing-demographic perspective, disregarding price value of the housing units.Nevertheless, although equation (4) is useful due to its simplicity, the intrinsic dynamicsof the system is more complicated. Each layer follows a different trend due to the integrationof the ODEs. In order to provide a vacancy rate for each of the layers, i.e. , the pace at whicheach layer becomes empty, we fit an exponential function of time to the predicted occupancylevels, such as o i ( t ) ∝ exp ( κ i t ), for 20 years since 1991. Note that o i ( t ) is not an exponentialdecay function, but since we are mostly interested in short periods of time (10 to 50 years),the exponential fits are adequate. As it can be seen in Fig. 4B, layers 2 and 3 undergo theheaviest vacancy rate at current market conditions, while the other layers are in a more stablesituation. This indicates that the middle class layers are subject to the heaviest speculativeprocess, while the other layers may have a lower demand. O cc upa t i on l e v e l κ ( y ea r − ) (B) BA Figure 4:
A. Occupation level for 4 different years. The occupation level has been obtained as the sumof paid property, property with mortgage, and property passed from parents to sons and daughters dividedby the total number of housing units. The solid line represents the prediction of the model by using 1991data. B. Decay rate of the occupation for each of the layers. This decay rate has been obtained by fittingthe o i ( t ) ∝ exp ( κ i t ) from 1991 top 2010. The economical conditions that may affect the transition probabilities among layers areincorporated into the model by means of a single parameter T , that plays a role similarto the temperature in statistical physics. For example, if the interest rates are raised, theprobability of transition from one layer to the rest or from the base pool to any layer isdecreased, which is modeled by a temperature decrement. Another example, if the RPIgrows excessively, then the temperature is also lowered, with the consequent decrease of thetransition probabilities. Unfortunately, we do not have data to quantify the dependence oftransition probabilities on economical conditions. To be able to estimate them we wouldneed data that indicates the number of mortgage requests and the amount loaned as a10unction of time. This has to be provided by banks and we currently do not have suchdata. The general dependence of the transition probability rates are therefore unknown.Initially, we assume that the transition rates ( µ ij and η i ) have a linear dependence on theparameter T . Therefore, the parameters µ ij and η i in equation 3 are replaced by T µ ij and T η i . This dependence basically implies that the economical conditions uniformly affect allthe probability rates in the same way. The goal is to determine whether there is a valueof the temperature parameter that leads to a sudden change on the occupancy levels. Aslong as the dependence of the probability rates on the temperature is an analytic function,the existence of critical behavior in the dynamical system should be preserved. In otherwords, if µ ij ( T ) and η i ( T ) are discontinuous, then the criticality emerges from parameterdependence on T , not from the dynamical system under consideration. In this work, we canonly concentrate on the dynamical system behavior.First, to determine the effects of freezing the system by temperature reduction we fit thedecay rate of the occupancy level to a exponential function as shown in the previous section(see Fig. 5A). The middle layers are more robust to temperature changes. The middle layersshow more resilience to critical economical conditions while both extremes of the multilayermodel are highly sensitive. This contrasts with the depletion resistance observed in Fig.4B for layer one. Thus, the lowest layer is the most sensitive to changes in the economicalconditions. κ ( y ea r − ) (A) O c upa t i on deg r ee L1L2L3L5 BA Figure 5: (A) Decay rate of the occupancy level for each of the layers. This decay rate has been obtained byfitting the o i ( t ) ∝ exp ( κ i t ). Circles represent current market conditions. Squares represent a hypotheticalsituation where the probability rates are reduced 10%. (B) Effect of a sudden decrease in the temperature in2005 using the parameter conditions estimated for 1991. Layer 1 is the most sensitive to changing economicalconditions, while layers 2 and 3 are nearly unaffected. Note that, for clarity reasons, we have not includedhigher layers because of their similarity to layer 5. As an example of the sensitivity of the lowest layer we introduce in 2005 a sudden decreasein the temperature that can be seen as a strong increase in the interest rates. In Fig. 5B wecan see that a sudden change in the temperature yields a sharp slope modification in layer1. On the other hand, it is striking to see that layers 2 and 3 are not as heavily impaired.In summary, using the 1991 market parameters and the nonlinear dynamics formal-ism proposed in this article, the housing market of Madrid is not subject to critical (non-11ontinuous) behavior on the temperature control parameter. It is clear that layers (2 and3) are very resilient to temperature changes. Nevertheless, although criticality is absent,layer 1 undergoes the most rapid changes to variable economical conditions. The layers withmost expensive houses can potentially undergo serious reversals under difficult economicalconditions. By means of this multilayer model we can identify which sectors of the housingpopulation would be more affected. An overall regulation of the market following equation(4) may not lead to uniform stabilization of the system. Multi-layer models can be used todetect the individual impact of global policies.
The temperature parameter is not a strong control parameter, since it does not lead the sys-tem into an asymptotic equilibrium. Therefore, the question of whether there is a parameterthat can globally keep the system in a nearly asymptotic equilibrium remains unanswered.A possible candidate is the growth rate of the number of houses, k . As we have seen insection 4 and fig. 2B the house growth rate is exponential for the city of Madrid, which is incontrast with the nearly flat increase of the base pool population (fig. 2C). This situation isnot sustainable, and the model can be useful to determine what to expect when regulatingthe house growth. We have integrated equations 3 with the parameter values obtained for1991, suddenly changing the growth factor k at 2005. Time to 10% depletion versus k isshown in figure 6. As we can see, there is a power law dependence. Basically, the timerequired to a 10% depletion is proportional to (1 /k ) (fits actually yield k − . to k − . ).This simple solution allows to easily estimate what the effects of house growth regulationwill have in the market.This housing regulation policy contradicts the intuition about a voiced general opinionthat the land in Madrid should be deregulated to build more housing units and, therefore,decrease the overall prices. This suggestion might reduce the prices of the housing units,yet it could worsen the current situation in Madrid in the long term. From the housing-demographic point of view, and according to this nonlinear model, more deregulation ofland can push the system even farther out of equilibrium.It is interesting to note that regulation policies for the lowest and the higher layers arecloser to each other than the middle ones. Policies to regulate the poorest layer will alsocontribute positively in regulating the upper ones. The middle layers follow a differentdynamics on its own mostly due to the fact that they receive and send family units fromboth sides of the distribution. As pointed out by Brian Arthur [Arthur 1999]:“...complexity economics, is not an adjunctto standard economic theory, but theory at a more general, out-of-equilibrium level.” In thispaper, we develop a novel multi-layered nonlinear dynamical framework for modeling the12 −4 −3 −2 K (year −1 )10 T i m e t o % dep l e t i on Figure 6:
Time to 10% depletion as a function of a sudden decrease of the housing unit growth factor in2005 using the parameter conditions estimated for 1991. The solid line corresponds to layer 1, the dottedline to layer 2, and the dashed line to layer 4. The rest of the layers follow the same power law. housing market dynamics. Using realistic data with the highest available precision, we showthat the housing market for the city of Madrid is currently driving away from equilibrium.This model can be used as a testing tool to determine the global effects of policy changesby governments. It is a tool to determine the effect of control parameters for all potentialtypes of dynamics even for out-of-equilibrium conditions. Traditional econometric toolsapproximate the dynamics near set points, which are estimated (sometimes believed) to bethe equilibrium points. When the system gets out of equilibrium there is not much guidanceabout what to expect, except waiting till it gets near the equilibrium point again. Generaltools to estimate effects of policy changes in the long run can be useful. Here we give anexample that is able to provide a good prediction of the global level of occupancy in Madridin 2001 (see Fig. 4A).Our original goal was to find out whether the current house-market conditions in Madridcould lead to a critical condition such that, while slowly moving a parameter value as thetemperature, the levels of occupancy suddenly drop to low levels. Fortunately, we did not findthe existence of such criticality in the context of this model that uses to the maximum possibleextent realistic data. When the interest rate is increased, which reduces the temperatureparameter in our model, the occupancy levels of the medium range housing are not seriouslyaffected. The first layer suffers dramatic consequences, and the wealthier layers undergoserious readjustments.Although this result appears to be good news (except for the poorest sector of the housingunits), the fact is that the city of Madrid is in a serious out-of-equilibrium condition thatasymptotically drives the levels of occupancy to 0. According to our model, interest ratecorrections will not modify this condition. The pragmatical way to control this situation13s to individually regulate the amount of new construction for each of the housing layers.Smooth changes in the construction rate can have a positive effect slowing down the out-of-equilibrium condition.
We want to thank Montserrat Mart´ınez and Cesar Pe˜nas for much of the data mining. Thiswork has been funded by Ministerio de Industria (Spain) PROFIT FIT 340000-2004-103.
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